Moment-problem formulation of a minimax quantization procedure

• Published in: Physical review. A, Atomic, molecular, and optical physics Atomic, molecular, and optical physics, 1994-08, Vol.50 (2), p.988-996
• Main Authors: Handy, CR, Appiah, K, Bessis, D
• Format: Article
• Language: English
• Subjects: General Physics; Physics
• ISSN: 1050-2947; 1094-1622
• DOI: 10.1103/PhysRevA.50.988

The Eigenvalue Moment Method (EMM) is a general theory for generating converging lower and upper bounds to the discrete, low-lying, spectrum of Schrödinger Hamiltonians. Recently, Handy, Giraud, and Bessis developed a dynamical systems EMM formulation through the discovery of a fundamental convex function $$ F_E[u] = Min_{\sigma =0,1} \left\langle V^{(\sigma ,E)}[u] \right\vert{\cal M}^{(\sigma ,E)}[u] \left\vert V^{(\sigma ,E)}[u] \right\rangle . $$ By incorporating this within the $ C $-{\sl Shift\/} EMM theory of Handy and Lee, there results a new Max Min quantization procedure involving the function $$ V(E) = Max_u Min_{\sigma =0,1} \left\langle V^{(\sigma ,E,S)}[u] \right\vert S^{-1}_\sigma{\cal M}^{(\sigma ,E)}[u]S^{-1}_\sigma \left\vert V^{(\sigma ,E,S)} \right\rangle , $$ whose local maxima converge to the discrete energy states of the system. We discuss the relevant theory and present several examples.